Contents
1 METRIC SPACES
1.1 Definition and Examples of Metric Spaces
1.2 Open Sets, Neighbourhoods and Interior Points
1.3 Accumulation Points, Closed Sets, Closure
1.4 Metric Subspaces
1.5 Equivalent Metrics
1.6 Continuous Functions On Metric Spaces
1.7 Complete and Separable Metric Spaces
1.8 Practice Problems
2 TOPOLOGICAL SPACES
2.1 Definition and Examples of Topological Spaces
2.2 Bases and Subspace Topologies
2.3 Continuous Functions
2.4 Homeomorphisms
2.5 Product Topologies
2.6 Hausdorff Spaces
2.7 Neighbourhoods, Accumulation Points, Interior Points, Boundary Points
2.8 Practice Problems
3 COMPACTNESS
3.1 Definition and Examples of Compact Spaces
3.2 The Heine-Borel Theorem
3.3 Practice Problems
4 CONNECTEDNESS
4.1 Definition and Examples of Connected Spaces
4.2 The Intermediate Value Theorem
4.3 Path-Connectedness
4.4 Practice Problems
1.1 Definition and Examples of Metric Spaces
1.2 Open Sets, Neighbourhoods and Interior Points
1.3 Accumulation Points, Closed Sets, Closure
1.4 Metric Subspaces
1.5 Equivalent Metrics
1.6 Continuous Functions On Metric Spaces
1.7 Complete and Separable Metric Spaces
1.8 Practice Problems
2 TOPOLOGICAL SPACES
2.1 Definition and Examples of Topological Spaces
2.2 Bases and Subspace Topologies
2.3 Continuous Functions
2.4 Homeomorphisms
2.5 Product Topologies
2.6 Hausdorff Spaces
2.7 Neighbourhoods, Accumulation Points, Interior Points, Boundary Points
2.8 Practice Problems
3 COMPACTNESS
3.1 Definition and Examples of Compact Spaces
3.2 The Heine-Borel Theorem
3.3 Practice Problems
4 CONNECTEDNESS
4.1 Definition and Examples of Connected Spaces
4.2 The Intermediate Value Theorem
4.3 Path-Connectedness
4.4 Practice Problems